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If N is contained in K and both are normal in G, then (G/N)/(K/N) is canonically isomorphic to G/K.
Third Isomorphism Theorem (Groups). Let be a group, and let be normal subgroups of . Then , and there is a canonical isomorphism of quotient groups
induced by .
The map , , is surjective with kernel , so the result follows from the first isomorphism theorem.
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Let be a group and let be a subgroup. The subgroup is normal in , written , if for every ,
Normality says that is stable under the conjugation action of on itself. Equivalently, is normal iff every left coset of equals the corresponding right coset, and this is exactly the hypothesis needed to form the quotient group .
Let be a normal subgroup of a group . The quotient group is the set of (left) cosets equipped with the operation
Let be a group homomorphism. Its kernel is a normal subgroup of , and induces a group isomorphism
In particular, if is surjective, then .